Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem

Stéphane Mischler (1) , Clément Mouhot (1) , Mariano Rodriguez Ricard (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 FACULTAD DE MATEMATICA Y COMPUTACION - Facultad de Matemática y Computación

We develop the Cauchy theory of the spatially homogeneous inelastic Boltzmann equation for hard spheres, for a general form of collision rate which includes in particular variable restitution coefficients depending on the kinetic energy and the relative velocity as well as the sticky particles model. We prove (local in time) non-concentration estimates in Orlicz spaces, from which we deduce weak stability and existence theorem. Strong stability together with uniqueness and instantaneous appearance of exponential moments are proved under additional smoothness assumption on the initial datum, for a restricted class of collision rates. Concerning the long-time behaviour, we give conditions for the cooling process to occur or not in finite time.

Mots clés: inelastic Boltzmann equation; hard spheres; variable restitution coefficient; Cauchy problem; Orlicz spaces; cooling process

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Stéphane Mischler, Clément Mouhot, Mariano Rodriguez Ricard. Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem. Journal of Statistical Physics, 2006, 124 (2-4), pp.655-702. ⟨10.1007/s10955-006-9096-9⟩. ⟨hal-00087232⟩